.TH  DSYSVX 1 "November 2006" " LAPACK driver routine (version 3.1) " " LAPACK driver routine (version 3.1) " 
.SH NAME
DSYSVX - the diagonal pivoting factorization to compute the solution to a real system of linear equations A * X = B,
.SH SYNOPSIS
.TP 19
SUBROUTINE DSYSVX(
FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV, B,
LDB, X, LDX, RCOND, FERR, BERR, WORK, LWORK,
IWORK, INFO )
.TP 19
.ti +4
CHARACTER
FACT, UPLO
.TP 19
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INTEGER
INFO, LDA, LDAF, LDB, LDX, LWORK, N, NRHS
.TP 19
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DOUBLE
PRECISION RCOND
.TP 19
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INTEGER
IPIV( * ), IWORK( * )
.TP 19
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DOUBLE
PRECISION A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
BERR( * ), FERR( * ), WORK( * ), X( LDX, * )
.SH PURPOSE
DSYSVX uses the diagonal pivoting factorization to compute the
solution to a real system of linear equations A * X = B,
where A is an N-by-N symmetric matrix and X and B are N-by-NRHS
matrices.
.br

Error bounds on the solution and a condition estimate are also
provided.
.br

.SH DESCRIPTION
The following steps are performed:
.br

1. If FACT = \(aqN\(aq, the diagonal pivoting method is used to factor A.
   The form of the factorization is
.br
      A = U * D * U**T,  if UPLO = \(aqU\(aq, or
.br
      A = L * D * L**T,  if UPLO = \(aqL\(aq,
.br
   where U (or L) is a product of permutation and unit upper (lower)
   triangular matrices, and D is symmetric and block diagonal with
   1-by-1 and 2-by-2 diagonal blocks.
.br

2. If some D(i,i)=0, so that D is exactly singular, then the routine
   returns with INFO = i. Otherwise, the factored form of A is used
   to estimate the condition number of the matrix A.  If the
   reciprocal of the condition number is less than machine precision,
   INFO = N+1 is returned as a warning, but the routine still goes on
   to solve for X and compute error bounds as described below.

3. The system of equations is solved for X using the factored form
   of A.
.br

4. Iterative refinement is applied to improve the computed solution
   matrix and calculate error bounds and backward error estimates
   for it.
.br

.SH ARGUMENTS
.TP 8
FACT    (input) CHARACTER*1
Specifies whether or not the factored form of A has been
supplied on entry.
= \(aqF\(aq:  On entry, AF and IPIV contain the factored form of
A.  AF and IPIV will not be modified.
= \(aqN\(aq:  The matrix A will be copied to AF and factored.
.TP 8
UPLO    (input) CHARACTER*1
.br
= \(aqU\(aq:  Upper triangle of A is stored;
.br
= \(aqL\(aq:  Lower triangle of A is stored.
.TP 8
N       (input) INTEGER
The number of linear equations, i.e., the order of the
matrix A.  N >= 0.
.TP 8
NRHS    (input) INTEGER
The number of right hand sides, i.e., the number of columns
of the matrices B and X.  NRHS >= 0.
.TP 8
A       (input) DOUBLE PRECISION array, dimension (LDA,N)
The symmetric matrix A.  If UPLO = \(aqU\(aq, the leading N-by-N
upper triangular part of A contains the upper triangular part
of the matrix A, and the strictly lower triangular part of A
is not referenced.  If UPLO = \(aqL\(aq, the leading N-by-N lower
triangular part of A contains the lower triangular part of
the matrix A, and the strictly upper triangular part of A is
not referenced.
.TP 8
LDA     (input) INTEGER
The leading dimension of the array A.  LDA >= max(1,N).
.TP 8
AF      (input or output) DOUBLE PRECISION array, dimension (LDAF,N)
If FACT = \(aqF\(aq, then AF is an input argument and on entry
contains the block diagonal matrix D and the multipliers used
to obtain the factor U or L from the factorization
A = U*D*U**T or A = L*D*L**T as computed by DSYTRF.

If FACT = \(aqN\(aq, then AF is an output argument and on exit
returns the block diagonal matrix D and the multipliers used
to obtain the factor U or L from the factorization
A = U*D*U**T or A = L*D*L**T.
.TP 8
LDAF    (input) INTEGER
The leading dimension of the array AF.  LDAF >= max(1,N).
.TP 8
IPIV    (input or output) INTEGER array, dimension (N)
If FACT = \(aqF\(aq, then IPIV is an input argument and on entry
contains details of the interchanges and the block structure
of D, as determined by DSYTRF.
If IPIV(k) > 0, then rows and columns k and IPIV(k) were
interchanged and D(k,k) is a 1-by-1 diagonal block.
If UPLO = \(aqU\(aq and IPIV(k) = IPIV(k-1) < 0, then rows and
columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
is a 2-by-2 diagonal block.  If UPLO = \(aqL\(aq and IPIV(k) =
IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were
interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.

If FACT = \(aqN\(aq, then IPIV is an output argument and on exit
contains details of the interchanges and the block structure
of D, as determined by DSYTRF.
.TP 8
B       (input) DOUBLE PRECISION array, dimension (LDB,NRHS)
The N-by-NRHS right hand side matrix B.
.TP 8
LDB     (input) INTEGER
The leading dimension of the array B.  LDB >= max(1,N).
.TP 8
X       (output) DOUBLE PRECISION array, dimension (LDX,NRHS)
If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X.
.TP 8
LDX     (input) INTEGER
The leading dimension of the array X.  LDX >= max(1,N).
.TP 8
RCOND   (output) DOUBLE PRECISION
The estimate of the reciprocal condition number of the matrix
A.  If RCOND is less than the machine precision (in
particular, if RCOND = 0), the matrix is singular to working
precision.  This condition is indicated by a return code of
INFO > 0.
.TP 8
FERR    (output) DOUBLE PRECISION array, dimension (NRHS)
The estimated forward error bound for each solution vector
X(j) (the j-th column of the solution matrix X).
If XTRUE is the true solution corresponding to X(j), FERR(j)
is an estimated upper bound for the magnitude of the largest
element in (X(j) - XTRUE) divided by the magnitude of the
largest element in X(j).  The estimate is as reliable as
the estimate for RCOND, and is almost always a slight
overestimate of the true error.
.TP 8
BERR    (output) DOUBLE PRECISION array, dimension (NRHS)
The componentwise relative backward error of each solution
vector X(j) (i.e., the smallest relative change in
any element of A or B that makes X(j) an exact solution).
.TP 8
WORK    (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
.TP 8
LWORK   (input) INTEGER
The length of WORK.  LWORK >= max(1,3*N), and for best
performance, when FACT = \(aqN\(aq, LWORK >= max(1,3*N,N*NB), where
NB is the optimal blocksize for DSYTRF.

If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.
.TP 8
IWORK   (workspace) INTEGER array, dimension (N)
.TP 8
INFO    (output) INTEGER
= 0: successful exit
.br
< 0: if INFO = -i, the i-th argument had an illegal value
.br
> 0: if INFO = i, and i is
.br
<= N:  D(i,i) is exactly zero.  The factorization
has been completed but the factor D is exactly
singular, so the solution and error bounds could
not be computed. RCOND = 0 is returned.
= N+1: D is nonsingular, but RCOND is less than machine
precision, meaning that the matrix is singular
to working precision.  Nevertheless, the
solution and error bounds are computed because
there are a number of situations where the
computed solution can be more accurate than the
value of RCOND would suggest.
